1. Field of the Invention
The present invention relates to a method and an apparatus for analyzing a twinned crystal with the use of X-ray single crystal structure analytical equipment using an X-ray diffraction method and for displaying the analytical result three-dimensionally.
2. Description of the Related Art
Single crystal structure analysis using the X-ray diffraction method requires high-accurate measurement of intensities of diffraction spots which are recorded in an X-ray detector. If a sample is a single crystal, it is easy to determine diffraction spot intensities with a high accuracy with the use of the ordinary X-ray single crystal structure analytical equipment. On the other hand, when a sample is a twinned crystal, especially when classified to non-merohedral twin, appearance of diffraction spots depends upon the geometrical relationship between the twin components, resulting in that, on the X-ray detector, diffraction spots coming from different twin components may overlap perfectly or partially each other. In this case, it is difficult to determine diffraction spot intensities with a high accuracy and therefore it is difficult to execute crystal structure analysis. In some cases, structural analysis would become impossible. Thus, the most important consideration in the analysis of the diffraction data on the X-ray detector is the degree of overlap of the diffraction spots on the X-ray detector. Accordingly, it is very important to understand the relationship between twin components and the degree of overlap of the reciprocal lattice points.
The term “twinned crystal” means a crystal made of two or more single crystal components joined with each other. It would be impossible in general to divide mechanically the twin crystal into its components. FIG. 17A is a perspective view of one example of the twinned crystal made of two twin components, and FIG. 17B is a perspective view of another example of the twinned crystal made of two twin components. When such a twinned crystal is studied using the structural analysis based on the X-ray diffraction method, especially when the crystal is classified to the non-merohedral twin, there appear two types of diffraction spots: one type has diffraction spots which come from reciprocal lattice spots overlap each other in reciprocal space and thus are not separable on the X-ray detector, and other type has diffraction spots which are separable on the X-ray detector. Under the circumstances, when X-ray diffraction intensities are measured and thereafter analyzed, it may become difficult to execute the structure analysis if the geometrical relationship between the twin components is not properly understood.
Some methods have been known for expressing the geometrical relationship between components consisting of a twinned crystal. The first method is to mathematically express the relationship between the twin components with the use of a three-row, three-column matrix. The second method is to display, in reciprocal space, the respective reciprocal lattice points coming from the twin components. FIGS. 18A to 18D show prior art expressions displaying two-dimensionally two nets of the reciprocal lattice points coming from two twin components, the two nets overlapping each other. Four kinds of expressions are shown depending upon the symmetrical type of the twinned crystal. FIGS. 18A and 18B show two types belonging to TLQS (Twin Lattice Quasi-Symmetry), while FIGS. 18C and 18D show other two types belonging to TLS (Twin Lattice Symmetry). FIG. 19 shows an expression displaying two-dimensionally the respective real-space unit lattices of two twin components in a two-dimensional plane. This twinned crystal has a specific relationship in which 90-degree rotation of the unit lattice (a1,b1) of the first twin component around c-axis allows it to coincide with the unit lattice (a2,b2) of the second twin component. FIG. 20 shows an expression displaying X-ray diffraction spots corresponding to the real-space unit lattices shown in FIG. 19 in a two-dimensional reciprocal lattice plane. X-ray diffraction spots coming from the first twin component are denoted by black circles, while X-ray diffraction spots coming from the second twin component are denoted by white circles.
The method of mathematically expressing the relationship between the twin components with the use of the three-row, three-column matrix is disclosed in TWINLAW and HKLF5, two programs for the handling of non-merohedral twins, Michael Bolte, Journal of Applied Crystallography, (2004). 37, 162-165. The methods of displaying the twinned crystal shown in FIGS. 18A to 18D and FIGS. 19 and 20 are disclosed in TWINNING WORKSHOP, AsCA/Crystal 23, Instructor: Victor G. Young, Jr., University of Minesota, 14:00-16:00, 11 Aug. 2003.
The above-described prior art methods for expressing the twinned crystal are inadequate to simply express the three-dimensional relationship between the twin components. The mathematical method using the matrix can define exactly the relationship between the twin components, but it is very difficult to understand the three-dimensional relationship between the twin components based on the matrix. The method of expressing the twinned crystal on the two-dimensional reciprocal lattice net as shown in FIGS. 18A to 18D and FIG. 20 would be useful, only for a scholar of crystallography having considerable knowledge and experience, to understand the three-dimensional relationship between the twin components.